Answer
$$5.333 $$
Work Step by Step
Given $$\lim _{\theta \rightarrow 0} \frac{\sin ^{2} 2 \theta-\theta \sin 4 \theta}{\theta^{4}}$$ Consider $$ f(\theta)= \frac{\sin ^{2} 2 \theta-\theta \sin 4 \theta}{\theta^{4}}$$ From the following figure, we can observe that \begin{align*} \lim _{\theta\rightarrow 0^+} f( \theta)&= 5.333\\ \lim _{\theta \rightarrow 0^-} f( \theta)&= 5.333 \end{align*} Then $$\lim _{\theta \rightarrow 0} \frac{\sin ^{2} 2 \theta-\theta \sin 4 \theta}{\theta^{4}}=5.333 $$
